New Zealand studies (see Table 7-3). One approach might be to apply the hierarchical analysis directly to the estimated BMDs, although the committee felt it appropriate to apply the analysis to the inverse BMDs instead. One advantage of working with the inverse BMDs is that very large and undefined values are transformed to zero. Working with the inverse BMDs also has some theoretical justification, because in the context of a linear model, the estimated BMD is simply a constant divided by the estimated dose-response slope (see Equation 7-1).
To describe the committee's approach in more detail, it is useful to define some notation. Let be the inverse of the BMD estimated for the jth outcome, j = 1, . . . Ji, within study, i = 1, . . . I. The corresponding standard errors, , can be estimated by subtracting from the inverse of the BMDL and then dividing by 1.64. The hierarchical model can be expressed as
where a, b, c, d, m, and n are chosen so that the priors are all relatively noninformative. In other words, we assume that the true inverse BMDs for each outcome are normally distributed around a study-specific mean value and that these study-specific values are in turn normally distributed around an overall mean. We fit the hierarchical model using the BUGS (Bayesian inference Using Gibbs Sampling) software package (Spiegelhalter et al. 1996). The product of the analysis is a series of simulated distributions of the various random variables defined in the model. Applying an inverse transformation again converts those results to yield estimates of the distribution of the quantities of interest, namely, BMDs. In addition to providing an estimate of distribution of true BMDs corresponding to different outcomes from different studies, the output from the program allows computation of so-called posterior estimates of the true BMDs, given the observed values. The advantage